Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quasirandom points and global function fields
FFA '95 Proceedings of the third international conference on Finite fields and applications
Monte Carlo Variance of Scrambled Net Quadrature
SIAM Journal on Numerical Analysis
A generalized discrepancy and quadrature error bound
Mathematics of Computation
Scrambling sobol' and niederreiter-xing points
Journal of Complexity
On the L2-discrepancy for anchored boxes
Journal of Complexity
Monte Carlo and Quasi-Monte Carlo Methods, 1998: Proceedings of a Conference Held at the Claremont Graduate University, Claremont, California, USA, JU
The Mean Square Discrepancy of Scrambled (t,s)-Sequences
SIAM Journal on Numerical Analysis
Algorithm 823: Implementing scrambled digital sequences
ACM Transactions on Mathematical Software (TOMS)
The distribution of the discrepancy of scrambled digital (t, m, s)-nets
Mathematics and Computers in Simulation - Special issue: 3rd IMACS seminar on Monte Carlo methods - MCM 2001
Algorithm 823: Implementing scrambled digital sequences
ACM Transactions on Mathematical Software (TOMS)
Journal of Complexity
I-binomial scrambling of digital nets and sequences
Journal of Complexity
Optimal quadrature for Haar wavelet spaces
Mathematics of Computation
On the mean square weighted L2 discrepancy of randomized digital nets in prime base
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Constructions of (t ,m,s)-nets and (t,s)-sequences
Finite Fields and Their Applications
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Digital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (λ, t, m, s)-net in base b with n = λbm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error.